Document Zbl 1400.90089

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Appl. Math.</a> <a href="/?q=in%3A414696" title="Articles in this Issue">78, No. 5, 2891-2917 (2018)</a>. </div> <div class="abstract">The aim of the article is to derive and investigate a discrete velocity model for traffic flow with a nonlinear hyperbolic part fulfilling some requirements. The model has to guarantee the sub characteristic condition and convergence to the scalar conservation law. In fact, by a particular choice of velocities, $v_1=0$, $v_2=1$ and denoting $\rho$ the density, $q$ the mean flux , $H$ a “measure for the look ahead and nonlocality of the equations”, the obtained macroscopic system reads ${\partial}_t\rho+\partial_xq=0$, ${\partial}_tq+((Hq)/(1-\rho)){\partial}_x\rho+ (1-(Hq)/(1-\rho)){\partial}_xq= -(1/\epsilon)(q-F(\rho))$. Investigated are questions related to boundary conditions for the scalar conservation law derived from boundary value problems to nonlinear kinetic relaxation system and the stability of the kinetic model near equilibrium and stable wave solutions. For $H=1$, a relaxation scheme is shown and additionally the case $H\to 0$ and the case without a relaxation term are discussed. The article ends with some numerical results.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A1582">Claudia Simionescu-Badea (Wien)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6964655">Cited in <strong>13</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B20" title="MSC2020">90B20</a> </td> <td class="space"> Traffic problems in operations research </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L02" title="MSC2020">35L02</a> </td> <td class="space"> First-order hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L04" title="MSC2020">35L04</a> </td> <td class="space"> Initial-boundary value problems for first-order hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L60" title="MSC2020">35L60</a> </td> <td class="space"> First-order nonlinear hyperbolic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35L65" title="MSC2020">35L65</a> </td> <td class="space"> Hyperbolic conservation laws </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adiscrete+velocity+model">discrete velocity model</a>; <a href="/?q=ut%3Atraffic+flow">traffic flow</a>; <a href="/?q=ut%3Arelaxation+system">relaxation system</a>; <a href="/?q=ut%3Acluster+dynamics">cluster dynamics</a>; <a href="/?q=ut%3Arelaxation+scheme">relaxation scheme</a>; <a href="/?q=ut%3Alinear+degenerate+hyperbolic+equations">linear degenerate hyperbolic equations</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1400.90089" data-ciurl="/ci/06964655" data-biburl="/bibtex/06964655.bib" data-amsurl="/amsrefs/06964655.bib" data-xmlurl="/xml/06964655.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06964655.pdf" title="Zbl 1400.90089 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1137/17M1152681" aria-label="DOI for “A nonlinear discrete velocity relaxation model for traffic flow”" title="10.1137/17M1152681">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1710.06262"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">D. 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